If \( A=\begin{bmatrix}\cos\alpha & \sin\alpha\\-\sin\alpha & \cos\alpha\end{bmatrix} \), then the value of \( 1000|A^{n}|+2|A| \) for a positive integer \( n \), is equal to
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 1002
\( \displaystyle |A|=\cos^2\alpha+\sin^2\alpha=1 \)
Since the determinant of a product is the product of the determinants, \( |A^n|=|A|^n \):
\( \displaystyle |A^n|=1^n=1 \)
\( \displaystyle 1000|A^n|+2|A|=1000(1)+2(1) \)
\( \displaystyle =1002 \)